Question

Asked Sep 23, 2019

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The mean of the commute time to work for a resident of a certain city is 28.3 minutes. Assume that the standard deviation of the commute time is 7.6 minutes to complete parts (a) through (c).

a). what minimum percentage of commuter in the city has a commute time within 2 standard deviations of the mean?

Step 1

A normal distribution can be visualized by looking at the curve below.

X-axis shows Z score which is the distance from mean in terms of standard deviation. A z score of 2 would mean that the distance of the data point from the mean is equal to 2 standard deviation.

Commute time within 2 standard deviations of the mean would mean all communte times between Z score of -2 and +2 as shown by the shaded portion under the curve.

Step 2

The area under the curve represents probability or percentage or proportion. Since the value of probability ranges from 0 to1, the total area under the curve is always equal to 1 or 100%.

To calculate area percentage or probability corresponding to any Z score , Z tables can be used which yields percentage of area to the left of a score

We are interested in P(-2<Z<2) which will be equal to P(Z<2) - P(Z<-2)

This can be represented graphically as shown below.

Step 3

Let's see how Z table can be read for a Z score of -2 and +2

To find P(Z<=-2) , we have to find the value that lies at the intersection of row -2 and column .00

(NOTE: If this was P(Z<=-1.57), then we would look for value at the intersec...

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